Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products:
\((a_1a_2)a_3=\sum \limits _{\sigma \in \mathbb {S}_3} x_{\sigma } a_{\sigma (1)}(a_{\sigma (2)}a_{\sigma (3)});\) 
such an operad is called a parametrized one-relation operad. For a particular choice of parameters \(\lbrace x_{\sigma }\rbrace \) , this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space \(V\)  is, as a graded vector space, isomorphic to the tensor algebra of \(V\) .
Bremner and Dotsenko classified, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, they proved that each such operad is isomorphic to one of the following five operads:
the left-nilpotent operad, the associative operad, the Leibniz operad, the Zinbiel operad, and the Poisson operad [1]}.
An algebra \(\bf A\)  is called a (left) Zinbiel algebra if it satisfies the identity
\((xy)z=x(yz+zy).\) 
Zinbiel algebras were introduced by Loday in [2]}.
Under the Koszul duality, the operad of Zinbiel algebras is dual to the operad of Leibniz algebras.
Zinbiel algebras are also known as pre-commutative algebras [3]}
and chronological algebras [4]}.
A Zinbiel algebra is equivalent to a commutative dendriform algebra [5]}.
It plays an important role in the definition of pre-Gerstenhaber algebras
[6]}.
The variety of Zinbiel algebras is a proper subvariety in the variety of right commutative algebras.
Each Zinbiel algebra with the commutator multiplication gives a Tortkara algebra [7]}, which has sprung up in unexpected areas of mathematics [8]}, [9]}.
Recently, the notion of matching Zinbiel algebras was introduced in [10]}.
Recently, Zinbiel algebras also appeared in a study of rack cohomology [11]},
number theory [12]} and
in a construction of a Cartesian differential category [13]}.
In recent years, there has been a strong interest in the study of Zinbiel algebras in the algebraic and the operad context
[14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [10]}, [21]}, [7]}, [23]}, [24]}, [25]}, [26]}.
